When inspecting the likelihood of [1]} and the assumptions of [2]}, we see the same independence assumptions for \(I_1, I_2, G\)  and \(T_1\) . Independence between \(I_1\)  and \(T_1\)  can also be found in [3]}. The variable \(C\)  can be understood as covariate and will be interpreted as the location of the infection. It can be used to capture heterogeneity in the infection dynamics: In a location with an increasing number of infections, it is more likely that the infection time \(T_1\)  lies towards the end of the infection window \([0,W]\)  conditionally on \((C,W)\) . In contrast, in locations with a low number of infections, \(T_1\)  is possibly uniformly distributed in \([0,W]\) . See Remark REF  below for further details and a motivation for certain choices of \(h\) . It has been reported (see e.g. [4]}) that the transmission dynamics, i.e., \(\varphi _I\)  and \(\varphi _G\) , can be different in different scenarios. Thus, we need to restrict to data coming from the same environment. Moreover, the above assumption means that we observe true transmission pairs (see e.g. [4]} for a parametric model which allows for an unknown number of intermediate infectors).
